All browser to my experience when you close them it logs out of sites your using and only when you log back in the login will be remembered and you choose the correct login and pswd to login to the site again. This is done to protect user and site security from security hacks on your computer and from malware. Some sites are more particular to logins that if you close their site they log you out for security reason and Google is one of them. So as long as your the only user and password protect computer to login to your computer no one else should have access to your account.

All browser to my experience when you close them it logs out of sites your using and only when you log back in the login will be remembered and you choose the correct login and pswd to login to the site again. This is done to protect user and site security from security hacks on your computer and from malware. Some sites are more particular to logins that if you close their site they log you out for security reason and Google is one of them. So as long as your the only user and password protect computer to login to your computer no one else should have access to your account.

Earlier today, before I cleared the cookies and cache and made the update, Firefox would keep my sessions logged in. Whenever I started Firefox, my email would already be logged in. This only changed now, hence why I find it strange

Detail like websites remembering you and log you in automatically is stored in cookies, so make sure you aren't clearing important cookies.

You can use these steps to make a website recognize and remember you.

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" to clear cookies keeps cookies with an allow exception
  in 102+ version toggling this setting makes changes to the "Clear history when Firefox closes" settings and those settings prevail
• using "Clear history when Firefox closes" in Firefox 102+ honors exceptions and keeps cookies with an allow exception, previous versions removed all cookies

Make sure to keep the "Site settings".

• clearing "Site settings" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions"
• Settings -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings

Detail like websites remembering you and log you in automatically is stored in cookies, so make sure you aren't clearing important cookies. You can use these steps to make a website recognize and remember you.

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" to clear cookies keeps cookies with an allow exception
  in 102+ version toggling this setting makes changes to the "Clear history when Firefox closes" settings and those settings prevail
• using "Clear history when Firefox closes" in Firefox 102+ honors exceptions and keeps cookies with an allow exception, previous versions removed all cookies

Make sure to keep the "Site settings".

• clearing "Site settings" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions"
• Settings -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings

I tried doing so (adding exceptions, etc) but this didn't work. Furthermore, I think I didn't have to do this before the recent changes. Google sites also never popped a window asking for permission for cookies, so I'm not sure Google is even trying to use cookies

You can try to create the cookie exception for https://google.com